By Fomin A.A.

Each τ -adic matrix represents either a quotient divisible crew and a torsion-free, finite-rank workforce. those representations are an equivalence and a duality of different types, respectively.

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2. Let X be a manifold, and n: E ~ X a mapping from some set E into X. Let {Ui } be an open covering of X, and for each i suppose that we are given a Banach space E and a bijection (commuting with the projection on Ui ), such that for each pair i, j and x E Ui n"0, the map (TjT t I) x is a 46 VECTOR BUNDLES [III, §l] top linear isomorphism, and condition VB 3 is satisfied as well as the cocycle condition. Then there exists a unique structure of manifold on E such that n is a morphism, such that Ti is an isomorphism making n into a vector bundle, and {( Ui , Ti)} into a trivialising covering.

Let f: U----F + ty lies in U for be a CP-morphism, and denote by y(P) the "vector" (y, ... , y) p times. Then the function DPf(x + ty) . y(P) is continuous in t, and we have f(x + y) = f(x) + + Df(x)y I! + ... + DP-If(x)y(p-I) (p _ I)! DPf(x + ty)y(P) dt. Jo (1(p-- tV-I I)! I Proof By the Hahn-Banach theorem, it suffices to show that both sides give the same thing when we apply a functional A (continuous linear map into R). 1, together with the known result when F = R. In this case, the proof proceeds by induction on p, and integration by parts, starting from f(x + y) - f(x) = J~ Df(x + ty)y dt.

Let f: X (i) (ii) ----+ Y be a CP-morphism. Let x EX. Then: f is an immersion at x if and only if there exists a chart (U, rp) at x and (V, t/J) at f(x) such that H u(rpx) is injective and splits. f is a submersion at x if and only if there exists a chart (U, rp) at x and (V, t/J) at f(x) such that H,u(rpx) is surjective and its kernel splits. 6 of the inverse mapping theorem. The conditions expressed in (i) and (ii) depend only on the derivative, and if they hold for one choice of charts (U, rp) and (V, t/J) respectively, then they hold for every choice of such charts.

### A category of matrices representing two categories of Abelian groups by Fomin A.A.

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